Questions tagged [topology]
In mathematics, topology examines the properties of space (such as connectedness and compactness) that are preserved under continuous deformations, such as stretching and bending, but not tearing or gluing.
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Find an example of a closed, achronal set $S$ in Minkowski spacetime such that $J^+(S)$ is not closed
This is one of the exercises on Wald's General Relativity:
Chapter 8, Problem 8.b Find an example of a closed, achronal set $S$ in Minkowski spacetime such that $J^+(S)$ is not closed. (Hint: ...
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Is a Klein-bottle-like topology allowed for GR?
As the spacetime of the universe seems to be quite flat, a torus topology comes mind easily. How about others?
Is it issue if manifold is non-orientable? I see challenges to find 3- or 4-Klein-bottle-...
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All Chern numbers as mapping degree?
The Chern number of Haldane model can be interpreted as a mapping degree from $T^2$ (1BZ) to $S^2$ (Bloch sphere). The question is whether all the Chern numbers can be interpreted in this way.
In ...
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Allowed Topologies for General Relativity
Studying the ADM formulation of General Relativity the ADM splitting comes out from the assumption that the spacetime is globally hyperbolic.
From that assumption thanks to Geroch's theorem, it is ...
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Topological proof of spin-statistics theorem confusion
I am currently studying the spin-statistics theorem. I have found a section on John Baez's website which presents a "proof" of the spin-statistics theorem. He states the theorem as:
This is ...
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Does the total Zak phase always sum to zero?
In 2D, the sum of the Chern numbers over all bands is zero. However, this result relies on the ability to define a Berry curvature, which is only possible in $d \geq 2$ dimensions. In 1D it is ...
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What is the topology of sine-Gordon equation?
In one pdf on solitons, I am finding the following written
For the sine-Gordon theory, it is much better to think of $\phi$ as a field modulo $2\pi$, i.e. as a function $\phi: R \rightarrow S_{1}$. ...
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Is a complex scalar field on a curved manifold $M$ just a complex-valued function on $M$?
I used to think of real and complex scalar fields as just functions, both on a flat and curved manifolds. Let's take a complex scalar field $\Phi$. In physics, usually we think of it as a map $\Phi: M ...
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Are non-trivial topologies in the gravitational path integral related to large gauge transformations in Yang-Mills?
While the gravitational path integral is not a well-understood concept mathematically, a number of works (particularly in recent research connected to AdS/CFT) emphasize the importance of integrating ...
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General element of the Poincare group
Consider the generators of the Poincare group are $M^{\mu\nu}$ and $P^\mu$. The Lorentz group elements are $g(\omega)\in SO(1,3)=e^{-\frac{i}{2}\omega_{\mu\nu}M^{\mu\nu}}$ and the translation group ...
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Global form of flavour symmetry groups in gauge theories
How do we work out the global nature of a flavour symmetry group? To be concrete, consider the simplest example of QED, preferably in D dimensions, with $N$ flavours of fermions with Lagrangian
$$\...
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A question about the topology of spacetime and the existence of CTCs
Let $(M, g)$ be a smooth Lorenzian time-oriented manifold.
Is it possible for the Lorenzian metric induced topology to be different from that of the manifold topology, without CTCs?
We know that the ...
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Single-valuedness of the monopole harmonics
It is known that it is impossible to have a non-singular vector potential for the monopole magnetic field.
How about the monopole harmonics?
Could the wave functions be made single-valued and finite ...
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GR metric tensor from extra dimensions
In a flat Euclidean 2d plane, the metric tensor is given by:
$g_{\mu\nu} = \begin{pmatrix}
1 & 0 \\
0 & 1
\end{pmatrix}$
If we imagine this plane as curving into an external 3rd dimension ...
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Why should a Cauchy surface be closed?
A Cauchy surface is defined on any spacetime $M$ as a subset $S$ which is closed, achronal, and whose domain of dependence $D(S) = M$.
Why do we include the "closed" condition in the above ...
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Getting rid of the theta term in the standard electroweak theory
This has already been asked here more than once, but the existing answers do not tackle my misunderstanding.
A topological $\theta$-term is understood to be physical, in the usual particle model ...
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Does the universe have an infinite volume? [duplicate]
The implications of a spatially infinite universe is profound, but so are the implications of a finite universe. What we know about this issue?
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Relation between Displacement Operator and Winding number
I am trying to implement a paper [https://arxiv.org/abs/2003.06086] using quantum computing techniques. In the supplementary material[SM] with the main paper, they introduce a displacement operator ...
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Understanding the shape of 2-dimensional orbifold given a metric tensor [closed]
I am trying to grasp some intuition about the shape represented by the following two-dimensional metric tensor:
$$ds^2=dr^2+n^2\cdot r^2d\theta^2 $$
where $r$ is "radial" coordinate, and $\...
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Non-Compactness in Penrose Singularity
I've been studying singularities in GR, and (obviously), came across PST.
Let us state it as the following:
Let $(M, g)$ be a connected globally hyperbolic
spacetime with a noncompact Cauchy ...
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Minkowski space and its compactified version: What's the difference?
Minkowski space has the metric $$\mathrm{d}s^2 = -\mathrm{d}t^2+\mathrm{d}x^2+\mathrm{d}y^2+\mathrm{d}z^2,$$
with all coordinates range from $-\infty$ to $+\infty$, and it's a solution to the Einstein ...
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Can anyons exist on a torus without any additional conditions?
While learning recently some more "advanced" stuff about path integral formalism I was introduced to the topological conditions that specify the process of construction of the propagator, i....
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Topology of Time
I came across the concept of topology of time and causality in Reichenbach book, "Philosophy of Space and Time". It would be nice to have list of references of recent developments of the ...
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Why must boosts be non-compact?
It is a common argument in the theory of kinematic groups (the groups of motions for a spacetime) that the subgroups generated by boosts must be non-compact[1][2][3]. This is true of all commonly used ...
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On what space of maps is Polyakov path integral actually defined?
This is a question more concerned about mathematical detail involving the Polyakov path integral.
In section $3.2$ of Polchinski's 1st String Theory book it is stated the following about Polyakov path ...
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When a curve is future (past) inextendible?
Future (past) endpoint: We say that $p\in M$ is a future (past) endpoint of a curve $\lambda$ if for every neighborhood $O$ of $p$ there exists a $t_0$ such that
$\lambda(t)\in O$ for all $t>t_0$ (...
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Question regarding the Theorem 8.1.2 (page 191) of Wald’s General Relativity book [duplicate]
Theorem 8.1.2: Let $(M, g_{ab})$ be an arbitrary spacetime, and $p\in M$. Then there exists a convex normal neighborhood of $p$, i.e., an open set $U$ with $p\in U$ such that for all $q,r\in U$ there ...
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Numerical calculation of Chern number for low energy continuum models
Can Fukui method be employed for the Chern number calculation in a low-energy continuum model? Let's consider the low-energy continuum version of the Kane-Mele model. If not what are some other ways ...
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Vortex and Anti-Vortex Pairs that form below the Berezinskii-Kosterlitz-Thouless transition and Abrikosov Vortices in superconductors
Please forgive me for my naivety of the subject, as I am relatively new and this is my first question as well. I am currently quite confused about Abrikosov vortices and Vortex and Anti-Vortex Pairs ...
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Characterising Minkowski spacetime as a flat manifold with some other property?
It is known that flat manifolds can be characterized as follows
If a pseudo-Riemannian manifold $M$ of signature $(s,t)$ has zero Riemann
curvature tensor everywhere on $M$, then the manifold is ...
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Is the electroweak gauge group a semidirect product?
In the typical treatment of electroweak theory, the gauge group is $G = \mathrm{SU}(2)_I \times \mathrm{U}(1)_Y$. This group is broken by the Higgs mechanism, while the combination of generators $Q = ...
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What does really mean to glue the endpoints of a closed string?
I'm almost all string theory standard textbooks such as Polchinski, Barton Zwiebach's book, etc. It is stated that the Worldsheet (or parameter space) flor the closed string is such that the points $(\...
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Self-connected Einstein-Rosen wormhole
The Einstein-Rosen wormhole is described by the metric
$$ \begin{equation}
\mathrm{d} s^{2}=\frac{u^{2}}{u^{2}+2 m} \mathrm{d} t^{2}-4\left(u^{2}+2 m\right) \mathrm{d} u^{2}-\left(u^{2}+2 m\right)^...
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Gauss-Bonnet term in tetrad formalism
In eq. 4 of https://arxiv.org/abs/hep-th/9508128 it is stated that the Gauss-Bonnet term for gravity in 4d can be written as
$$
S=\frac{1}{4}\int d^{4}x\,\epsilon^{\mu\nu\alpha\beta}\epsilon_{abcd}R_{\...
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Analogue of Bargmann's theorem for Super Lie groups
Bargmann's theorem gives the criteria under which a projective representation of a Lie group $G$ can be lifted to a representation of its universal cover. More generally, if this criterion, namely $H^...
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Why don’t black rings exist in 3+1 dimensions?
In higher dimensions black holes can take different forms. Besides spheroids, there are also “black rings” which have a toroidal event horizon. However, in our 3+1 dimension no black ring solution has ...
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Holes in phase space because of collision potential?
Consider a gas. Now, we already know the particle's duration of a collision is very small and it immediately bounces away from there. Can these be thought of as topological holes in one's phase space ...
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How is a wormhole (Einstein-Rosen Bridge) different than a tunnel?
What is the difference between an Einstein-Rosen Bridge (wormhole) and a tunnel through a mountain? Obviously, light that travelled around the mountain would take longer to reach other side so that ...
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Gauge transformation as transition functions?
I understand that transition functions are between different patches $U_i, U_j$, i.e. in $U_i\cap U_j$ you need a transition function to stitch local trivializations.
On the other hand, gauge ...
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Wormholes as instantons?
Are all wormholes gravitational instantons in the context of General Relativity?
My question concerns also the topology of spacetime in such case.
A full Wick rotation of the metric, seems to change ...
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In what way are Lie groups generated by the basis of their Lie algebra?
In this question, the answer by twistor59 says
by using the exponential map on linear combinations of [the lie algebra basis vectors], you generate (at least locally) a copy of the Lie group.
I'm ...
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Invariance of the Euler characteristic of a manifold with boundaries
It is known that in a compact manifold of dimension $d=4$, the following integral is invariant under small deformations of the metric tensor
$$
\int_\mathcal M \sqrt{g}(R^2-4R_{\mu\nu}R^{\mu\nu}+R_{\...
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How is magnetic reconnection possible, aren't field lines closed loops?
When I try to understand magnetic reconnection, I have a fundamental conflict with my intuition of magnetic fields.
I think it is literally a case of cognitive dissonance, my intuition and any kind of ...
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Why/When can we separate spacetime into space and time?
As far as I understand, for all practical applications in GR, we would need a way to split space and time. Since, often in practical applications and understanding physical phenomena, lengths and time ...
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Classical systems with compact phase space
In the Hamiltonian formalism of classical mechanics, a system with configuration space $Q$ is represented by a symplectic manifold $(T^*Q,\omega^\mathrm{can})$ called the phase space. The dynamics are ...
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Is it actually true that Chern-Simons theory is topological?
Chern-Simons theory has action $$\tag{1} S = \frac{k}{4\pi}\int_X tr(A\wedge dA + \frac{2}{3}A\wedge A\wedge A).$$
Here, $X$ is some compact 3-manifold, perhaps with boundary, and $A$ is a connection ...
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Tangent Space of a one-dimensional manifold
Is the tangent space of a one-dimensional smooth manifold always trivial, i.e. is $TM \cong M \times R$?
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How do you calculate the partition function on a manifold-with-corners in extended TQFT?
In Atiyah's formulation, a Topological Quantum Field Theory (TQFT), is a functor $Z:d\text{Bord}\to\text{Hilb}$. That is, $Z$ assigns:
\begin{align}
\text{Closed compact $(d-1)$-manifolds} &\to \...
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Do the Klein bottle and torus topologies break the Lorentz invariance?
According to this preprint, it seems that there are topologies (like the Klein bottle and the torus) that break some symmetries (like the Lorentz and translation invariances).
Is this right? Can they ...
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What is the rotational symmetry of the theorized spin-2 graviton?
We know and have actually measured in the lab with self-interference neutron experiments the 4π-symmetry (720° rotation Dirac Belt characteristic) of all spin-1/2 particles (except the neutrinos) thus ...
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